LMFDB - Embedded newform 8512.2.a.ch.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.9686622005$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 10x^{5} + 31x^{4} + 12x^{3} - 45x^{2} - 15x + 8 $ x^7 - 3x^6 - 10x^5 + 31x^4 + 12x^3 - 45x^2 - 15x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4256)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.77715$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77715 q^{3} +0.595310 q^{5} -1.00000 q^{7} +0.158273 q^{9} +O(q^{10})$	$q-1.77715 q^{3} +0.595310 q^{5} -1.00000 q^{7} +0.158273 q^{9} -6.33689 q^{11} -1.33561 q^{13} -1.05796 q^{15} +6.14658 q^{17} +1.00000 q^{19} +1.77715 q^{21} +0.680950 q^{23} -4.64561 q^{25} +5.05018 q^{27} -0.562766 q^{29} -7.08767 q^{31} +11.2616 q^{33} -0.595310 q^{35} -3.86658 q^{37} +2.37358 q^{39} +5.17937 q^{41} -6.54753 q^{43} +0.0942213 q^{45} +5.37612 q^{47} +1.00000 q^{49} -10.9234 q^{51} -11.0005 q^{53} -3.77241 q^{55} -1.77715 q^{57} -6.32544 q^{59} +11.2504 q^{61} -0.158273 q^{63} -0.795101 q^{65} -10.3344 q^{67} -1.21015 q^{69} -12.8814 q^{71} -4.12066 q^{73} +8.25595 q^{75} +6.33689 q^{77} -6.21320 q^{79} -9.44977 q^{81} -3.14945 q^{83} +3.65912 q^{85} +1.00012 q^{87} -1.33389 q^{89} +1.33561 q^{91} +12.5959 q^{93} +0.595310 q^{95} -2.06937 q^{97} -1.00296 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9}+O(q^{10}) $ 7 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9	$ 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9} - 3 q^{11} + 16 q^{13} + 8 q^{15} + 8 q^{17} + 7 q^{19} + 3 q^{21} + 10 q^{23} + 8 q^{25} - 6 q^{27} + 11 q^{29} + 14 q^{31} + 3 q^{33} - 5 q^{35} + 13 q^{37} - 6 q^{39} + 11 q^{41} - 11 q^{43} - 4 q^{45} + 7 q^{47} + 7 q^{49} - 24 q^{51} + 9 q^{53} - 8 q^{55} - 3 q^{57} - 23 q^{59} + 23 q^{61} - 8 q^{63} + 40 q^{65} - 16 q^{67} + 10 q^{69} + 3 q^{71} + 4 q^{73} - 48 q^{75} + 3 q^{77} + 29 q^{79} + 23 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} + 15 q^{89} - 16 q^{91} - 42 q^{93} + 5 q^{95} - 13 q^{97} - 47 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9 - 3 * q^11 + 16 * q^13 + 8 * q^15 + 8 * q^17 + 7 * q^19 + 3 * q^21 + 10 * q^23 + 8 * q^25 - 6 * q^27 + 11 * q^29 + 14 * q^31 + 3 * q^33 - 5 * q^35 + 13 * q^37 - 6 * q^39 + 11 * q^41 - 11 * q^43 - 4 * q^45 + 7 * q^47 + 7 * q^49 - 24 * q^51 + 9 * q^53 - 8 * q^55 - 3 * q^57 - 23 * q^59 + 23 * q^61 - 8 * q^63 + 40 * q^65 - 16 * q^67 + 10 * q^69 + 3 * q^71 + 4 * q^73 - 48 * q^75 + 3 * q^77 + 29 * q^79 + 23 * q^81 - 6 * q^83 + 6 * q^85 - 6 * q^87 + 15 * q^89 - 16 * q^91 - 42 * q^93 + 5 * q^95 - 13 * q^97 - 47 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.77715	−1.02604	−0.513020	−	0.858377i	$-0.671473\pi$
$3$	−1.77715	−1.02604	−0.513020	+	0.858377i	$0.671473\pi$
$4$	0	0
$5$	0.595310	0.266231	0.133115	−	0.991101i	$-0.457502\pi$
$5$	0.595310	0.266231	0.133115	+	0.991101i	$0.457502\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.158273	0.0527575
$10$	0	0
$11$	−6.33689	−1.91064	−0.955322	−	0.295569i	$-0.904491\pi$
$11$	−6.33689	−1.91064	−0.955322	+	0.295569i	$0.904491\pi$
$12$	0	0
$13$	−1.33561	−0.370431	−0.185215	−	0.982698i	$-0.559298\pi$
$13$	−1.33561	−0.370431	−0.185215	+	0.982698i	$0.559298\pi$
$14$	0	0
$15$	−1.05796	−0.273163
$16$	0	0
$17$	6.14658	1.49077	0.745383	−	0.666637i	$-0.232267\pi$
$17$	6.14658	1.49077	0.745383	+	0.666637i	$0.232267\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.77715	0.387807
$22$	0	0
$23$	0.680950	0.141988	0.0709940	−	0.997477i	$-0.477383\pi$
$23$	0.680950	0.141988	0.0709940	+	0.997477i	$0.477383\pi$
$24$	0	0
$25$	−4.64561	−0.929121
$26$	0	0
$27$	5.05018	0.971908
$28$	0	0
$29$	−0.562766	−0.104503	−0.0522515	−	0.998634i	$-0.516640\pi$
$29$	−0.562766	−0.104503	−0.0522515	+	0.998634i	$0.516640\pi$
$30$	0	0
$31$	−7.08767	−1.27298	−0.636491	−	0.771284i	$-0.719615\pi$
$31$	−7.08767	−1.27298	−0.636491	+	0.771284i	$0.719615\pi$
$32$	0	0
$33$	11.2616	1.96040
$34$	0	0
$35$	−0.595310	−0.100626
$36$	0	0
$37$	−3.86658	−0.635661	−0.317831	−	0.948148i	$-0.602954\pi$
$37$	−3.86658	−0.635661	−0.317831	+	0.948148i	$0.602954\pi$
$38$	0	0
$39$	2.37358	0.380077
$40$	0	0
$41$	5.17937	0.808882	0.404441	−	0.914564i	$-0.367466\pi$
$41$	5.17937	0.808882	0.404441	+	0.914564i	$0.367466\pi$
$42$	0	0
$43$	−6.54753	−0.998488	−0.499244	−	0.866461i	$-0.666389\pi$
$43$	−6.54753	−0.998488	−0.499244	+	0.866461i	$0.666389\pi$
$44$	0	0
$45$	0.0942213	0.0140457
$46$	0	0
$47$	5.37612	0.784188	0.392094	−	0.919925i	$-0.371751\pi$
$47$	5.37612	0.784188	0.392094	+	0.919925i	$0.371751\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.9234	−1.52958
$52$	0	0
$53$	−11.0005	−1.51104	−0.755519	−	0.655127i	$-0.772615\pi$
$53$	−11.0005	−1.51104	−0.755519	+	0.655127i	$0.772615\pi$
$54$	0	0
$55$	−3.77241	−0.508672
$56$	0	0
$57$	−1.77715	−0.235390
$58$	0	0
$59$	−6.32544	−0.823502	−0.411751	−	0.911296i	$-0.635083\pi$
$59$	−6.32544	−0.823502	−0.411751	+	0.911296i	$0.635083\pi$
$60$	0	0
$61$	11.2504	1.44046	0.720231	−	0.693735i	$-0.244036\pi$
$61$	11.2504	1.44046	0.720231	+	0.693735i	$0.244036\pi$
$62$	0	0
$63$	−0.158273	−0.0199405
$64$	0	0
$65$	−0.795101	−0.0986201
$66$	0	0
$67$	−10.3344	−1.26255	−0.631275	−	0.775559i	$-0.717468\pi$
$67$	−10.3344	−1.26255	−0.631275	+	0.775559i	$0.717468\pi$
$68$	0	0
$69$	−1.21015	−0.145685
$70$	0	0
$71$	−12.8814	−1.52874	−0.764369	−	0.644779i	$-0.776949\pi$
$71$	−12.8814	−1.52874	−0.764369	+	0.644779i	$0.776949\pi$
$72$	0	0
$73$	−4.12066	−0.482286	−0.241143	−	0.970490i	$-0.577522\pi$
$73$	−4.12066	−0.482286	−0.241143	+	0.970490i	$0.577522\pi$
$74$	0	0
$75$	8.25595	0.953315
$76$	0	0
$77$	6.33689	0.722155
$78$	0	0
$79$	−6.21320	−0.699040	−0.349520	−	0.936929i	$-0.613655\pi$
$79$	−6.21320	−0.699040	−0.349520	+	0.936929i	$0.613655\pi$
$80$	0	0
$81$	−9.44977	−1.04997
$82$	0	0
$83$	−3.14945	−0.345697	−0.172849	−	0.984948i	$-0.555297\pi$
$83$	−3.14945	−0.345697	−0.172849	+	0.984948i	$0.555297\pi$
$84$	0	0
$85$	3.65912	0.396888
$86$	0	0
$87$	1.00012	0.107224
$88$	0	0
$89$	−1.33389	−0.141392	−0.0706958	−	0.997498i	$-0.522522\pi$
$89$	−1.33389	−0.141392	−0.0706958	+	0.997498i	$0.522522\pi$
$90$	0	0
$91$	1.33561	0.140010
$92$	0	0
$93$	12.5959	1.30613
$94$	0	0
$95$	0.595310	0.0610775
$96$	0	0
$97$	−2.06937	−0.210112	−0.105056	−	0.994466i	$-0.533502\pi$
$97$	−2.06937	−0.210112	−0.105056	+	0.994466i	$0.533502\pi$
$98$	0	0
$99$	−1.00296	−0.100801
$100$	0	0
$101$	−16.8695	−1.67857	−0.839287	−	0.543689i	$-0.817027\pi$
$101$	−16.8695	−1.67857	−0.839287	+	0.543689i	$0.817027\pi$
$102$	0	0
$103$	11.5712	1.14015	0.570073	−	0.821594i	$-0.306915\pi$
$103$	11.5712	1.14015	0.570073	+	0.821594i	$0.306915\pi$
$104$	0	0
$105$	1.05796	0.103246
$106$	0	0
$107$	−7.41689	−0.717018	−0.358509	−	0.933526i	$-0.616715\pi$
$107$	−7.41689	−0.717018	−0.358509	+	0.933526i	$0.616715\pi$
$108$	0	0
$109$	14.5007	1.38892	0.694460	−	0.719531i	$-0.255643\pi$
$109$	14.5007	1.38892	0.694460	+	0.719531i	$0.255643\pi$
$110$	0	0
$111$	6.87150	0.652214
$112$	0	0
$113$	7.48371	0.704008	0.352004	−	0.935998i	$-0.385500\pi$
$113$	7.48371	0.704008	0.352004	+	0.935998i	$0.385500\pi$
$114$	0	0
$115$	0.405377	0.0378016
$116$	0	0
$117$	−0.211390	−0.0195430
$118$	0	0
$119$	−6.14658	−0.563456
$120$	0	0
$121$	29.1561	2.65056
$122$	0	0
$123$	−9.20454	−0.829945
$124$	0	0
$125$	−5.74213	−0.513591
$126$	0	0
$127$	4.90057	0.434855	0.217428	−	0.976076i	$-0.430233\pi$
$127$	4.90057	0.434855	0.217428	+	0.976076i	$0.430233\pi$
$128$	0	0
$129$	11.6360	1.02449
$130$	0	0
$131$	−10.4275	−0.911052	−0.455526	−	0.890222i	$-0.650549\pi$
$131$	−10.4275	−0.911052	−0.455526	+	0.890222i	$0.650549\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	3.00643	0.258752
$136$	0	0
$137$	−6.65890	−0.568908	−0.284454	−	0.958690i	$-0.591812\pi$
$137$	−6.65890	−0.568908	−0.284454	+	0.958690i	$0.591812\pi$
$138$	0	0
$139$	19.6735	1.66869	0.834344	−	0.551245i	$-0.185847\pi$
$139$	19.6735	1.66869	0.834344	+	0.551245i	$0.185847\pi$
$140$	0	0
$141$	−9.55420	−0.804608
$142$	0	0
$143$	8.46359	0.707761
$144$	0	0
$145$	−0.335020	−0.0278219
$146$	0	0
$147$	−1.77715	−0.146577
$148$	0	0
$149$	−8.20059	−0.671819	−0.335909	−	0.941894i	$-0.609044\pi$
$149$	−8.20059	−0.671819	−0.335909	+	0.941894i	$0.609044\pi$
$150$	0	0
$151$	19.3023	1.57080	0.785400	−	0.618989i	$-0.212457\pi$
$151$	19.3023	1.57080	0.785400	+	0.618989i	$0.212457\pi$
$152$	0	0
$153$	0.972836	0.0786491
$154$	0	0
$155$	−4.21936	−0.338907
$156$	0	0
$157$	10.9159	0.871185	0.435592	−	0.900144i	$-0.356539\pi$
$157$	10.9159	0.871185	0.435592	+	0.900144i	$0.356539\pi$
$158$	0	0
$159$	19.5496	1.55039
$160$	0	0
$161$	−0.680950	−0.0536664
$162$	0	0
$163$	−25.0706	−1.96368	−0.981839	−	0.189718i	$-0.939243\pi$
$163$	−25.0706	−1.96368	−0.981839	+	0.189718i	$0.939243\pi$
$164$	0	0
$165$	6.70415	0.521918
$166$	0	0
$167$	9.00507	0.696833	0.348416	−	0.937340i	$-0.386720\pi$
$167$	9.00507	0.696833	0.348416	+	0.937340i	$0.386720\pi$
$168$	0	0
$169$	−11.2162	−0.862781
$170$	0	0
$171$	0.158273	0.0121034
$172$	0	0
$173$	−2.53305	−0.192584	−0.0962921	−	0.995353i	$-0.530698\pi$
$173$	−2.53305	−0.192584	−0.0962921	+	0.995353i	$0.530698\pi$
$174$	0	0
$175$	4.64561	0.351175
$176$	0	0
$177$	11.2413	0.844946
$178$	0	0
$179$	22.9777	1.71743	0.858717	−	0.512450i	$-0.171262\pi$
$179$	22.9777	1.71743	0.858717	+	0.512450i	$0.171262\pi$
$180$	0	0
$181$	11.2375	0.835275	0.417638	−	0.908614i	$-0.362858\pi$
$181$	11.2375	0.835275	0.417638	+	0.908614i	$0.362858\pi$
$182$	0	0
$183$	−19.9936	−1.47797
$184$	0	0
$185$	−2.30181	−0.169233
$186$	0	0
$187$	−38.9502	−2.84832
$188$	0	0
$189$	−5.05018	−0.367347
$190$	0	0
$191$	3.80480	0.275306	0.137653	−	0.990481i	$-0.456044\pi$
$191$	3.80480	0.275306	0.137653	+	0.990481i	$0.456044\pi$
$192$	0	0
$193$	5.61590	0.404241	0.202121	−	0.979361i	$-0.435217\pi$
$193$	5.61590	0.404241	0.202121	+	0.979361i	$0.435217\pi$
$194$	0	0
$195$	1.41302	0.101188
$196$	0	0
$197$	−1.91133	−0.136176	−0.0680882	−	0.997679i	$-0.521690\pi$
$197$	−1.91133	−0.136176	−0.0680882	+	0.997679i	$0.521690\pi$
$198$	0	0
$199$	23.5608	1.67018	0.835092	−	0.550110i	$-0.185414\pi$
$199$	23.5608	1.67018	0.835092	+	0.550110i	$0.185414\pi$
$200$	0	0
$201$	18.3658	1.29543
$202$	0	0
$203$	0.562766	0.0394984
$204$	0	0
$205$	3.08333	0.215349
$206$	0	0
$207$	0.107776	0.00749094
$208$	0	0
$209$	−6.33689	−0.438332
$210$	0	0
$211$	17.6602	1.21578	0.607889	−	0.794022i	$-0.292016\pi$
$211$	17.6602	1.21578	0.607889	+	0.794022i	$0.292016\pi$
$212$	0	0
$213$	22.8922	1.56855
$214$	0	0
$215$	−3.89781	−0.265828
$216$	0	0
$217$	7.08767	0.481142
$218$	0	0
$219$	7.32304	0.494845
$220$	0	0
$221$	−8.20942	−0.552226
$222$	0	0
$223$	−13.3804	−0.896018	−0.448009	−	0.894029i	$-0.647867\pi$
$223$	−13.3804	−0.896018	−0.448009	+	0.894029i	$0.647867\pi$
$224$	0	0
$225$	−0.735272	−0.0490181
$226$	0	0
$227$	−26.4815	−1.75764	−0.878819	−	0.477155i	$-0.841668\pi$
$227$	−26.4815	−1.75764	−0.878819	+	0.477155i	$0.841668\pi$
$228$	0	0
$229$	9.11208	0.602144	0.301072	−	0.953601i	$-0.402656\pi$
$229$	9.11208	0.602144	0.301072	+	0.953601i	$0.402656\pi$
$230$	0	0
$231$	−11.2616	−0.740960
$232$	0	0
$233$	10.0041	0.655394	0.327697	−	0.944783i	$-0.393728\pi$
$233$	10.0041	0.655394	0.327697	+	0.944783i	$0.393728\pi$
$234$	0	0
$235$	3.20046	0.208775
$236$	0	0
$237$	11.0418	0.717243
$238$	0	0
$239$	19.6648	1.27201	0.636005	−	0.771685i	$-0.280586\pi$
$239$	19.6648	1.27201	0.636005	+	0.771685i	$0.280586\pi$
$240$	0	0
$241$	−19.4622	−1.25367	−0.626836	−	0.779152i	$-0.715650\pi$
$241$	−19.4622	−1.25367	−0.626836	+	0.779152i	$0.715650\pi$
$242$	0	0
$243$	1.64313	0.105407
$244$	0	0
$245$	0.595310	0.0380330
$246$	0	0
$247$	−1.33561	−0.0849827
$248$	0	0
$249$	5.59706	0.354699
$250$	0	0
$251$	7.85031	0.495507	0.247754	−	0.968823i	$-0.420308\pi$
$251$	7.85031	0.495507	0.247754	+	0.968823i	$0.420308\pi$
$252$	0	0
$253$	−4.31511	−0.271288
$254$	0	0
$255$	−6.50282	−0.407223
$256$	0	0
$257$	−2.70348	−0.168638	−0.0843192	−	0.996439i	$-0.526872\pi$
$257$	−2.70348	−0.168638	−0.0843192	+	0.996439i	$0.526872\pi$
$258$	0	0
$259$	3.86658	0.240257
$260$	0	0
$261$	−0.0890704	−0.00551332
$262$	0	0
$263$	6.19081	0.381742	0.190871	−	0.981615i	$-0.438869\pi$
$263$	6.19081	0.381742	0.190871	+	0.981615i	$0.438869\pi$
$264$	0	0
$265$	−6.54872	−0.402285
$266$	0	0
$267$	2.37052	0.145073
$268$	0	0
$269$	8.83762	0.538839	0.269420	−	0.963023i	$-0.413168\pi$
$269$	8.83762	0.538839	0.269420	+	0.963023i	$0.413168\pi$
$270$	0	0
$271$	−7.48139	−0.454462	−0.227231	−	0.973841i	$-0.572967\pi$
$271$	−7.48139	−0.454462	−0.227231	+	0.973841i	$0.572967\pi$
$272$	0	0
$273$	−2.37358	−0.143656
$274$	0	0
$275$	29.4387	1.77522
$276$	0	0
$277$	−1.74162	−0.104644	−0.0523219	−	0.998630i	$-0.516662\pi$
$277$	−1.74162	−0.104644	−0.0523219	+	0.998630i	$0.516662\pi$
$278$	0	0
$279$	−1.12178	−0.0671594
$280$	0	0
$281$	5.37920	0.320896	0.160448	−	0.987044i	$-0.448706\pi$
$281$	5.37920	0.320896	0.160448	+	0.987044i	$0.448706\pi$
$282$	0	0
$283$	−17.0268	−1.01214	−0.506068	−	0.862493i	$-0.668902\pi$
$283$	−17.0268	−1.01214	−0.506068	+	0.862493i	$0.668902\pi$
$284$	0	0
$285$	−1.05796	−0.0626680
$286$	0	0
$287$	−5.17937	−0.305729
$288$	0	0
$289$	20.7805	1.22238
$290$	0	0
$291$	3.67758	0.215584
$292$	0	0
$293$	16.0674	0.938666	0.469333	−	0.883021i	$-0.344494\pi$
$293$	16.0674	0.938666	0.469333	+	0.883021i	$0.344494\pi$
$294$	0	0
$295$	−3.76560	−0.219242
$296$	0	0
$297$	−32.0024	−1.85697
$298$	0	0
$299$	−0.909483	−0.0525967
$300$	0	0
$301$	6.54753	0.377393
$302$	0	0
$303$	29.9796	1.72228
$304$	0	0
$305$	6.69746	0.383495
$306$	0	0
$307$	−1.35504	−0.0773362	−0.0386681	−	0.999252i	$-0.512312\pi$
$307$	−1.35504	−0.0773362	−0.0386681	+	0.999252i	$0.512312\pi$
$308$	0	0
$309$	−20.5638	−1.16984
$310$	0	0
$311$	1.34852	0.0764674	0.0382337	−	0.999269i	$-0.487827\pi$
$311$	1.34852	0.0764674	0.0382337	+	0.999269i	$0.487827\pi$
$312$	0	0
$313$	11.1999	0.633055	0.316528	−	0.948583i	$-0.397483\pi$
$313$	11.1999	0.633055	0.316528	+	0.948583i	$0.397483\pi$
$314$	0	0
$315$	−0.0942213	−0.00530877
$316$	0	0
$317$	29.9972	1.68481	0.842405	−	0.538844i	$-0.181139\pi$
$317$	29.9972	1.68481	0.842405	+	0.538844i	$0.181139\pi$
$318$	0	0
$319$	3.56618	0.199668
$320$	0	0
$321$	13.1809	0.735689
$322$	0	0
$323$	6.14658	0.342005
$324$	0	0
$325$	6.20471	0.344175
$326$	0	0
$327$	−25.7700	−1.42509
$328$	0	0
$329$	−5.37612	−0.296395
$330$	0	0
$331$	−36.2557	−1.99279	−0.996397	−	0.0848067i	$-0.972973\pi$
$331$	−36.2557	−1.99279	−0.996397	+	0.0848067i	$0.972973\pi$
$332$	0	0
$333$	−0.611973	−0.0335359
$334$	0	0
$335$	−6.15218	−0.336130
$336$	0	0
$337$	−6.06317	−0.330282	−0.165141	−	0.986270i	$-0.552808\pi$
$337$	−6.06317	−0.330282	−0.165141	+	0.986270i	$0.552808\pi$
$338$	0	0
$339$	−13.2997	−0.722341
$340$	0	0
$341$	44.9137	2.43222
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.720416	−0.0387859
$346$	0	0
$347$	12.7388	0.683854	0.341927	−	0.939727i	$-0.388920\pi$
$347$	12.7388	0.683854	0.341927	+	0.939727i	$0.388920\pi$
$348$	0	0
$349$	11.9316	0.638684	0.319342	−	0.947640i	$-0.396538\pi$
$349$	11.9316	0.638684	0.319342	+	0.947640i	$0.396538\pi$
$350$	0	0
$351$	−6.74506	−0.360025
$352$	0	0
$353$	−11.2660	−0.599631	−0.299816	−	0.953997i	$-0.596925\pi$
$353$	−11.2660	−0.599631	−0.299816	+	0.953997i	$0.596925\pi$
$354$	0	0
$355$	−7.66842	−0.406997
$356$	0	0
$357$	10.9234	0.578129
$358$	0	0
$359$	17.0796	0.901426	0.450713	−	0.892669i	$-0.351170\pi$
$359$	17.0796	0.901426	0.450713	+	0.892669i	$0.351170\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−51.8149	−2.71958
$364$	0	0
$365$	−2.45307	−0.128399
$366$	0	0
$367$	−8.20498	−0.428297	−0.214148	−	0.976801i	$-0.568698\pi$
$367$	−8.20498	−0.428297	−0.214148	+	0.976801i	$0.568698\pi$
$368$	0	0
$369$	0.819753	0.0426746
$370$	0	0
$371$	11.0005	0.571119
$372$	0	0
$373$	−0.815800	−0.0422405	−0.0211203	−	0.999777i	$-0.506723\pi$
$373$	−0.815800	−0.0422405	−0.0211203	+	0.999777i	$0.506723\pi$
$374$	0	0
$375$	10.2046	0.526965
$376$	0	0
$377$	0.751634	0.0387111
$378$	0	0
$379$	−18.5580	−0.953260	−0.476630	−	0.879104i	$-0.658142\pi$
$379$	−18.5580	−0.953260	−0.476630	+	0.879104i	$0.658142\pi$
$380$	0	0
$381$	−8.70906	−0.446179
$382$	0	0
$383$	−9.28157	−0.474266	−0.237133	−	0.971477i	$-0.576208\pi$
$383$	−9.28157	−0.474266	−0.237133	+	0.971477i	$0.576208\pi$
$384$	0	0
$385$	3.77241	0.192260
$386$	0	0
$387$	−1.03629	−0.0526778
$388$	0	0
$389$	13.4543	0.682162	0.341081	−	0.940034i	$-0.389207\pi$
$389$	13.4543	0.682162	0.341081	+	0.940034i	$0.389207\pi$
$390$	0	0
$391$	4.18552	0.211671
$392$	0	0
$393$	18.5312	0.934776
$394$	0	0
$395$	−3.69878	−0.186106
$396$	0	0
$397$	36.7449	1.84417	0.922087	−	0.386983i	$-0.126483\pi$
$397$	36.7449	1.84417	0.922087	+	0.386983i	$0.126483\pi$
$398$	0	0
$399$	1.77715	0.0889689
$400$	0	0
$401$	5.94622	0.296940	0.148470	−	0.988917i	$-0.452565\pi$
$401$	5.94622	0.296940	0.148470	+	0.988917i	$0.452565\pi$
$402$	0	0
$403$	9.46634	0.471552
$404$	0	0
$405$	−5.62554	−0.279535
$406$	0	0
$407$	24.5021	1.21452
$408$	0	0
$409$	−26.0707	−1.28911	−0.644556	−	0.764557i	$-0.722958\pi$
$409$	−26.0707	−1.28911	−0.644556	+	0.764557i	$0.722958\pi$
$410$	0	0
$411$	11.8339	0.583722
$412$	0	0
$413$	6.32544	0.311255
$414$	0	0
$415$	−1.87490	−0.0920353
$416$	0	0
$417$	−34.9629	−1.71214
$418$	0	0
$419$	−4.09313	−0.199963	−0.0999813	−	0.994989i	$-0.531878\pi$
$419$	−4.09313	−0.199963	−0.0999813	+	0.994989i	$0.531878\pi$
$420$	0	0
$421$	29.4390	1.43477	0.717385	−	0.696677i	$-0.245339\pi$
$421$	29.4390	1.43477	0.717385	+	0.696677i	$0.245339\pi$
$422$	0	0
$423$	0.850893	0.0413719
$424$	0	0
$425$	−28.5546	−1.38510
$426$	0	0
$427$	−11.2504	−0.544443
$428$	0	0
$429$	−15.0411	−0.726191
$430$	0	0
$431$	−24.6970	−1.18961	−0.594806	−	0.803869i	$-0.702771\pi$
$431$	−24.6970	−1.18961	−0.594806	+	0.803869i	$0.702771\pi$
$432$	0	0
$433$	14.2377	0.684221	0.342111	−	0.939660i	$-0.388858\pi$
$433$	14.2377	0.684221	0.342111	+	0.939660i	$0.388858\pi$
$434$	0	0
$435$	0.595382	0.0285464
$436$	0	0
$437$	0.680950	0.0325743
$438$	0	0
$439$	25.8059	1.23165	0.615824	−	0.787883i	$-0.288823\pi$
$439$	25.8059	1.23165	0.615824	+	0.787883i	$0.288823\pi$
$440$	0	0
$441$	0.158273	0.00753679
$442$	0	0
$443$	18.0465	0.857414	0.428707	−	0.903444i	$-0.358969\pi$
$443$	18.0465	0.857414	0.428707	+	0.903444i	$0.358969\pi$
$444$	0	0
$445$	−0.794076	−0.0376428
$446$	0	0
$447$	14.5737	0.689313
$448$	0	0
$449$	−20.4175	−0.963563	−0.481782	−	0.876291i	$-0.660010\pi$
$449$	−20.4175	−0.963563	−0.481782	+	0.876291i	$0.660010\pi$
$450$	0	0
$451$	−32.8211	−1.54549
$452$	0	0
$453$	−34.3032	−1.61170
$454$	0	0
$455$	0.795101	0.0372749
$456$	0	0
$457$	−26.9470	−1.26053	−0.630263	−	0.776382i	$-0.717053\pi$
$457$	−26.9470	−1.26053	−0.630263	+	0.776382i	$0.717053\pi$
$458$	0	0
$459$	31.0414	1.44889
$460$	0	0
$461$	−37.1541	−1.73044	−0.865219	−	0.501395i	$-0.832820\pi$
$461$	−37.1541	−1.73044	−0.865219	+	0.501395i	$0.832820\pi$
$462$	0	0
$463$	−29.9505	−1.39192	−0.695958	−	0.718082i	$-0.745020\pi$
$463$	−29.9505	−1.39192	−0.695958	+	0.718082i	$0.745020\pi$
$464$	0	0
$465$	7.49845	0.347732
$466$	0	0
$467$	34.5022	1.59657	0.798285	−	0.602280i	$-0.205741\pi$
$467$	34.5022	1.59657	0.798285	+	0.602280i	$0.205741\pi$
$468$	0	0
$469$	10.3344	0.477199
$470$	0	0
$471$	−19.3992	−0.893870
$472$	0	0
$473$	41.4909	1.90775
$474$	0	0
$475$	−4.64561	−0.213155
$476$	0	0
$477$	−1.74108	−0.0797187
$478$	0	0
$479$	17.4697	0.798213	0.399107	−	0.916905i	$-0.369320\pi$
$479$	17.4697	0.798213	0.399107	+	0.916905i	$0.369320\pi$
$480$	0	0
$481$	5.16423	0.235469
$482$	0	0
$483$	1.21015	0.0550639
$484$	0	0
$485$	−1.23192	−0.0559384
$486$	0	0
$487$	−20.3957	−0.924216	−0.462108	−	0.886824i	$-0.652907\pi$
$487$	−20.3957	−0.924216	−0.462108	+	0.886824i	$0.652907\pi$
$488$	0	0
$489$	44.5542	2.01481
$490$	0	0
$491$	−6.05808	−0.273397	−0.136699	−	0.990613i	$-0.543649\pi$
$491$	−6.05808	−0.273397	−0.136699	+	0.990613i	$0.543649\pi$
$492$	0	0
$493$	−3.45909	−0.155789
$494$	0	0
$495$	−0.597070	−0.0268363
$496$	0	0
$497$	12.8814	0.577809
$498$	0	0
$499$	3.43566	0.153801	0.0769006	−	0.997039i	$-0.475498\pi$
$499$	3.43566	0.153801	0.0769006	+	0.997039i	$0.475498\pi$
$500$	0	0
$501$	−16.0034	−0.714978
$502$	0	0
$503$	−9.44472	−0.421119	−0.210560	−	0.977581i	$-0.567529\pi$
$503$	−9.44472	−0.421119	−0.210560	+	0.977581i	$0.567529\pi$
$504$	0	0
$505$	−10.0426	−0.446888
$506$	0	0
$507$	19.9328	0.885248
$508$	0	0
$509$	19.4717	0.863066	0.431533	−	0.902097i	$-0.357973\pi$
$509$	19.4717	0.863066	0.431533	+	0.902097i	$0.357973\pi$
$510$	0	0
$511$	4.12066	0.182287
$512$	0	0
$513$	5.05018	0.222971
$514$	0	0
$515$	6.88846	0.303542
$516$	0	0
$517$	−34.0679	−1.49830
$518$	0	0
$519$	4.50162	0.197599
$520$	0	0
$521$	7.70152	0.337410	0.168705	−	0.985667i	$-0.446042\pi$
$521$	7.70152	0.337410	0.168705	+	0.985667i	$0.446042\pi$
$522$	0	0
$523$	20.1150	0.879566	0.439783	−	0.898104i	$-0.355055\pi$
$523$	20.1150	0.879566	0.439783	+	0.898104i	$0.355055\pi$
$524$	0	0
$525$	−8.25595	−0.360319
$526$	0	0
$527$	−43.5649	−1.89772
$528$	0	0
$529$	−22.5363	−0.979839
$530$	0	0
$531$	−1.00114	−0.0434460
$532$	0	0
$533$	−6.91761	−0.299635
$534$	0	0
$535$	−4.41535	−0.190892
$536$	0	0
$537$	−40.8349	−1.76216
$538$	0	0
$539$	−6.33689	−0.272949
$540$	0	0
$541$	−24.4852	−1.05270	−0.526350	−	0.850268i	$-0.676440\pi$
$541$	−24.4852	−1.05270	−0.526350	+	0.850268i	$0.676440\pi$
$542$	0	0
$543$	−19.9707	−0.857026
$544$	0	0
$545$	8.63244	0.369773
$546$	0	0
$547$	27.8603	1.19122	0.595610	−	0.803274i	$-0.296910\pi$
$547$	27.8603	1.19122	0.595610	+	0.803274i	$0.296910\pi$
$548$	0	0
$549$	1.78062	0.0759952
$550$	0	0
$551$	−0.562766	−0.0239746
$552$	0	0
$553$	6.21320	0.264212
$554$	0	0
$555$	4.09067	0.173639
$556$	0	0
$557$	36.8766	1.56251	0.781255	−	0.624212i	$-0.214580\pi$
$557$	36.8766	1.56251	0.781255	+	0.624212i	$0.214580\pi$
$558$	0	0
$559$	8.74493	0.369871
$560$	0	0
$561$	69.2205	2.92249
$562$	0	0
$563$	43.4402	1.83078	0.915392	−	0.402563i	$-0.131881\pi$
$563$	43.4402	1.83078	0.915392	+	0.402563i	$0.131881\pi$
$564$	0	0
$565$	4.45513	0.187429
$566$	0	0
$567$	9.44977	0.396853
$568$	0	0
$569$	17.9279	0.751577	0.375788	−	0.926705i	$-0.377372\pi$
$569$	17.9279	0.751577	0.375788	+	0.926705i	$0.377372\pi$
$570$	0	0
$571$	14.1579	0.592492	0.296246	−	0.955112i	$-0.404265\pi$
$571$	14.1579	0.592492	0.296246	+	0.955112i	$0.404265\pi$
$572$	0	0
$573$	−6.76171	−0.282475
$574$	0	0
$575$	−3.16343	−0.131924
$576$	0	0
$577$	40.7278	1.69552	0.847761	−	0.530379i	$-0.177950\pi$
$577$	40.7278	1.69552	0.847761	+	0.530379i	$0.177950\pi$
$578$	0	0
$579$	−9.98031	−0.414767
$580$	0	0
$581$	3.14945	0.130661
$582$	0	0
$583$	69.7091	2.88705
$584$	0	0
$585$	−0.125843	−0.00520296
$586$	0	0
$587$	19.9962	0.825333	0.412667	−	0.910882i	$-0.364597\pi$
$587$	19.9962	0.825333	0.412667	+	0.910882i	$0.364597\pi$
$588$	0	0
$589$	−7.08767	−0.292042
$590$	0	0
$591$	3.39672	0.139722
$592$	0	0
$593$	22.7027	0.932287	0.466143	−	0.884709i	$-0.345643\pi$
$593$	22.7027	0.932287	0.466143	+	0.884709i	$0.345643\pi$
$594$	0	0
$595$	−3.65912	−0.150009
$596$	0	0
$597$	−41.8712	−1.71368
$598$	0	0
$599$	17.4338	0.712327	0.356163	−	0.934424i	$-0.384085\pi$
$599$	17.4338	0.712327	0.356163	+	0.934424i	$0.384085\pi$
$600$	0	0
$601$	−15.9884	−0.652182	−0.326091	−	0.945338i	$-0.605732\pi$
$601$	−15.9884	−0.652182	−0.326091	+	0.945338i	$0.605732\pi$
$602$	0	0
$603$	−1.63566	−0.0666090
$604$	0	0
$605$	17.3569	0.705660
$606$	0	0
$607$	14.1982	0.576288	0.288144	−	0.957587i	$-0.406962\pi$
$607$	14.1982	0.576288	0.288144	+	0.957587i	$0.406962\pi$
$608$	0	0
$609$	−1.00012	−0.0405270
$610$	0	0
$611$	−7.18039	−0.290488
$612$	0	0
$613$	−47.9776	−1.93780	−0.968898	−	0.247462i	$-0.920403\pi$
$613$	−47.9776	−1.93780	−0.968898	+	0.247462i	$0.920403\pi$
$614$	0	0
$615$	−5.47956	−0.220957
$616$	0	0
$617$	−12.4820	−0.502507	−0.251254	−	0.967921i	$-0.580843\pi$
$617$	−12.4820	−0.502507	−0.251254	+	0.967921i	$0.580843\pi$
$618$	0	0
$619$	−20.7684	−0.834752	−0.417376	−	0.908734i	$-0.637050\pi$
$619$	−20.7684	−0.834752	−0.417376	+	0.908734i	$0.637050\pi$
$620$	0	0
$621$	3.43893	0.137999
$622$	0	0
$623$	1.33389	0.0534410
$624$	0	0
$625$	19.8097	0.792387
$626$	0	0
$627$	11.2616	0.449746
$628$	0	0
$629$	−23.7662	−0.947622
$630$	0	0
$631$	4.31988	0.171972	0.0859858	−	0.996296i	$-0.472596\pi$
$631$	4.31988	0.171972	0.0859858	+	0.996296i	$0.472596\pi$
$632$	0	0
$633$	−31.3849	−1.24744
$634$	0	0
$635$	2.91736	0.115772
$636$	0	0
$637$	−1.33561	−0.0529187
$638$	0	0
$639$	−2.03877	−0.0806525
$640$	0	0
$641$	−25.0506	−0.989438	−0.494719	−	0.869053i	$-0.664729\pi$
$641$	−25.0506	−0.989438	−0.494719	+	0.869053i	$0.664729\pi$
$642$	0	0
$643$	21.8163	0.860352	0.430176	−	0.902745i	$-0.358452\pi$
$643$	21.8163	0.860352	0.430176	+	0.902745i	$0.358452\pi$
$644$	0	0
$645$	6.92700	0.272750
$646$	0	0
$647$	48.6618	1.91309	0.956547	−	0.291579i	$-0.0941806\pi$
$647$	48.6618	1.91309	0.956547	+	0.291579i	$0.0941806\pi$
$648$	0	0
$649$	40.0836	1.57342
$650$	0	0
$651$	−12.5959	−0.493671
$652$	0	0
$653$	38.5650	1.50917	0.754583	−	0.656205i	$-0.227839\pi$
$653$	38.5650	1.50917	0.754583	+	0.656205i	$0.227839\pi$
$654$	0	0
$655$	−6.20758	−0.242550
$656$	0	0
$657$	−0.652187	−0.0254442
$658$	0	0
$659$	19.8133	0.771817	0.385909	−	0.922537i	$-0.373888\pi$
$659$	19.8133	0.771817	0.385909	+	0.922537i	$0.373888\pi$
$660$	0	0
$661$	20.9889	0.816372	0.408186	−	0.912899i	$-0.366161\pi$
$661$	20.9889	0.816372	0.408186	+	0.912899i	$0.366161\pi$
$662$	0	0
$663$	14.5894	0.566605
$664$	0	0
$665$	−0.595310	−0.0230851
$666$	0	0
$667$	−0.383216	−0.0148382
$668$	0	0
$669$	23.7790	0.919350
$670$	0	0
$671$	−71.2923	−2.75221
$672$	0	0
$673$	36.9137	1.42292	0.711460	−	0.702727i	$-0.248034\pi$
$673$	36.9137	1.42292	0.711460	+	0.702727i	$0.248034\pi$
$674$	0	0
$675$	−23.4612	−0.903021
$676$	0	0
$677$	30.1339	1.15814	0.579070	−	0.815278i	$-0.303416\pi$
$677$	30.1339	1.15814	0.579070	+	0.815278i	$0.303416\pi$
$678$	0	0
$679$	2.06937	0.0794150
$680$	0	0
$681$	47.0617	1.80341
$682$	0	0
$683$	5.35906	0.205059	0.102529	−	0.994730i	$-0.467306\pi$
$683$	5.35906	0.205059	0.102529	+	0.994730i	$0.467306\pi$
$684$	0	0
$685$	−3.96411	−0.151461
$686$	0	0
$687$	−16.1936	−0.617823
$688$	0	0
$689$	14.6924	0.559735
$690$	0	0
$691$	40.4255	1.53786	0.768929	−	0.639334i	$-0.220790\pi$
$691$	40.4255	1.53786	0.768929	+	0.639334i	$0.220790\pi$
$692$	0	0
$693$	1.00296	0.0380991
$694$	0	0
$695$	11.7119	0.444256
$696$	0	0
$697$	31.8355	1.20585
$698$	0	0
$699$	−17.7789	−0.672460
$700$	0	0
$701$	15.2489	0.575945	0.287972	−	0.957639i	$-0.407019\pi$
$701$	15.2489	0.575945	0.287972	+	0.957639i	$0.407019\pi$
$702$	0	0
$703$	−3.86658	−0.145831
$704$	0	0
$705$	−5.68771	−0.214212
$706$	0	0
$707$	16.8695	0.634441
$708$	0	0
$709$	−18.0504	−0.677897	−0.338948	−	0.940805i	$-0.610071\pi$
$709$	−18.0504	−0.677897	−0.338948	+	0.940805i	$0.610071\pi$
$710$	0	0
$711$	−0.983380	−0.0368796
$712$	0	0
$713$	−4.82635	−0.180748
$714$	0	0
$715$	5.03846	0.188428
$716$	0	0
$717$	−34.9474	−1.30513
$718$	0	0
$719$	−12.8380	−0.478778	−0.239389	−	0.970924i	$-0.576947\pi$
$719$	−12.8380	−0.478778	−0.239389	+	0.970924i	$0.576947\pi$
$720$	0	0
$721$	−11.5712	−0.430935
$722$	0	0
$723$	34.5873	1.28632
$724$	0	0
$725$	2.61439	0.0970959
$726$	0	0
$727$	23.5441	0.873201	0.436601	−	0.899655i	$-0.356182\pi$
$727$	23.5441	0.873201	0.436601	+	0.899655i	$0.356182\pi$
$728$	0	0
$729$	25.4292	0.941823
$730$	0	0
$731$	−40.2449	−1.48851
$732$	0	0
$733$	26.0506	0.962202	0.481101	−	0.876665i	$-0.340237\pi$
$733$	26.0506	0.962202	0.481101	+	0.876665i	$0.340237\pi$
$734$	0	0
$735$	−1.05796	−0.0390233
$736$	0	0
$737$	65.4880	2.41228
$738$	0	0
$739$	42.8828	1.57747	0.788735	−	0.614734i	$-0.210737\pi$
$739$	42.8828	1.57747	0.788735	+	0.614734i	$0.210737\pi$
$740$	0	0
$741$	2.37358	0.0871956
$742$	0	0
$743$	−10.9844	−0.402977	−0.201489	−	0.979491i	$-0.564578\pi$
$743$	−10.9844	−0.402977	−0.201489	+	0.979491i	$0.564578\pi$
$744$	0	0
$745$	−4.88190	−0.178859
$746$	0	0
$747$	−0.498472	−0.0182381
$748$	0	0
$749$	7.41689	0.271007
$750$	0	0
$751$	−18.7476	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$751$	−18.7476	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$752$	0	0
$753$	−13.9512	−0.508410
$754$	0	0
$755$	11.4909	0.418195
$756$	0	0
$757$	4.34718	0.158001	0.0790006	−	0.996875i	$-0.474827\pi$
$757$	4.34718	0.158001	0.0790006	+	0.996875i	$0.474827\pi$
$758$	0	0
$759$	7.66860	0.278353
$760$	0	0
$761$	−14.6210	−0.530012	−0.265006	−	0.964247i	$-0.585374\pi$
$761$	−14.6210	−0.530012	−0.265006	+	0.964247i	$0.585374\pi$
$762$	0	0
$763$	−14.5007	−0.524962
$764$	0	0
$765$	0.579139	0.0209388
$766$	0	0
$767$	8.44831	0.305051
$768$	0	0
$769$	−46.7009	−1.68408	−0.842039	−	0.539417i	$-0.818645\pi$
$769$	−46.7009	−1.68408	−0.842039	+	0.539417i	$0.818645\pi$
$770$	0	0
$771$	4.80449	0.173030
$772$	0	0
$773$	29.9149	1.07596	0.537982	−	0.842956i	$-0.319187\pi$
$773$	29.9149	1.07596	0.537982	+	0.842956i	$0.319187\pi$
$774$	0	0
$775$	32.9265	1.18276
$776$	0	0
$777$	−6.87150	−0.246514
$778$	0	0
$779$	5.17937	0.185570
$780$	0	0
$781$	81.6278	2.92087
$782$	0	0
$783$	−2.84207	−0.101567
$784$	0	0
$785$	6.49835	0.231936
$786$	0	0
$787$	−1.45641	−0.0519154	−0.0259577	−	0.999663i	$-0.508264\pi$
$787$	−1.45641	−0.0519154	−0.0259577	+	0.999663i	$0.508264\pi$
$788$	0	0
$789$	−11.0020	−0.391682
$790$	0	0
$791$	−7.48371	−0.266090
$792$	0	0
$793$	−15.0261	−0.533591
$794$	0	0
$795$	11.6381	0.412760
$796$	0	0
$797$	12.0572	0.427089	0.213544	−	0.976933i	$-0.431499\pi$
$797$	12.0572	0.427089	0.213544	+	0.976933i	$0.431499\pi$
$798$	0	0
$799$	33.0448	1.16904
$800$	0	0
$801$	−0.211118	−0.00745947
$802$	0	0
$803$	26.1121	0.921477
$804$	0	0
$805$	−0.405377	−0.0142877
$806$	0	0
$807$	−15.7058	−0.552871
$808$	0	0
$809$	0.172092	0.00605042	0.00302521	−	0.999995i	$-0.499037\pi$
$809$	0.172092	0.00605042	0.00302521	+	0.999995i	$0.499037\pi$
$810$	0	0
$811$	−41.8400	−1.46920	−0.734600	−	0.678500i	$-0.762630\pi$
$811$	−41.8400	−1.46920	−0.734600	+	0.678500i	$0.762630\pi$
$812$	0	0
$813$	13.2956	0.466296
$814$	0	0
$815$	−14.9248	−0.522791
$816$	0	0
$817$	−6.54753	−0.229069
$818$	0	0
$819$	0.211390	0.00738657
$820$	0	0
$821$	−17.0138	−0.593786	−0.296893	−	0.954911i	$-0.595951\pi$
$821$	−17.0138	−0.593786	−0.296893	+	0.954911i	$0.595951\pi$
$822$	0	0
$823$	−18.5165	−0.645446	−0.322723	−	0.946493i	$-0.604598\pi$
$823$	−18.5165	−0.645446	−0.322723	+	0.946493i	$0.604598\pi$
$824$	0	0
$825$	−52.3170	−1.82145
$826$	0	0
$827$	12.6891	0.441244	0.220622	−	0.975359i	$-0.429191\pi$
$827$	12.6891	0.441244	0.220622	+	0.975359i	$0.429191\pi$
$828$	0	0
$829$	−16.0508	−0.557468	−0.278734	−	0.960368i	$-0.589915\pi$
$829$	−16.0508	−0.557468	−0.278734	+	0.960368i	$0.589915\pi$
$830$	0	0
$831$	3.09512	0.107369
$832$	0	0
$833$	6.14658	0.212966
$834$	0	0
$835$	5.36081	0.185518
$836$	0	0
$837$	−35.7940	−1.23722
$838$	0	0
$839$	−24.5786	−0.848547	−0.424274	−	0.905534i	$-0.639471\pi$
$839$	−24.5786	−0.848547	−0.424274	+	0.905534i	$0.639471\pi$
$840$	0	0
$841$	−28.6833	−0.989079
$842$	0	0
$843$	−9.55966	−0.329252
$844$	0	0
$845$	−6.67709	−0.229699
$846$	0	0
$847$	−29.1561	−1.00182
$848$	0	0
$849$	30.2592	1.03849
$850$	0	0
$851$	−2.63295	−0.0902562
$852$	0	0
$853$	−32.3755	−1.10852	−0.554258	−	0.832345i	$-0.686998\pi$
$853$	−32.3755	−1.10852	−0.554258	+	0.832345i	$0.686998\pi$
$854$	0	0
$855$	0.0942213	0.00322230
$856$	0	0
$857$	−8.68521	−0.296681	−0.148341	−	0.988936i	$-0.547393\pi$
$857$	−8.68521	−0.296681	−0.148341	+	0.988936i	$0.547393\pi$
$858$	0	0
$859$	42.5806	1.45283	0.726415	−	0.687256i	$-0.241185\pi$
$859$	42.5806	1.45283	0.726415	+	0.687256i	$0.241185\pi$
$860$	0	0
$861$	9.20454	0.313690
$862$	0	0
$863$	17.8080	0.606190	0.303095	−	0.952960i	$-0.401980\pi$
$863$	17.8080	0.606190	0.303095	+	0.952960i	$0.401980\pi$
$864$	0	0
$865$	−1.50795	−0.0512719
$866$	0	0
$867$	−36.9301	−1.25421
$868$	0	0
$869$	39.3724	1.33562
$870$	0	0
$871$	13.8027	0.467688
$872$	0	0
$873$	−0.327524	−0.0110850
$874$	0	0
$875$	5.74213	0.194119
$876$	0	0
$877$	−52.7393	−1.78088	−0.890440	−	0.455100i	$-0.849604\pi$
$877$	−52.7393	−1.78088	−0.890440	+	0.455100i	$0.849604\pi$
$878$	0	0
$879$	−28.5542	−0.963109
$880$	0	0
$881$	16.3578	0.551107	0.275553	−	0.961286i	$-0.411139\pi$
$881$	16.3578	0.551107	0.275553	+	0.961286i	$0.411139\pi$
$882$	0	0
$883$	7.36524	0.247860	0.123930	−	0.992291i	$-0.460450\pi$
$883$	7.36524	0.247860	0.123930	+	0.992291i	$0.460450\pi$
$884$	0	0
$885$	6.69205	0.224951
$886$	0	0
$887$	−16.9509	−0.569155	−0.284578	−	0.958653i	$-0.591853\pi$
$887$	−16.9509	−0.569155	−0.284578	+	0.958653i	$0.591853\pi$
$888$	0	0
$889$	−4.90057	−0.164360
$890$	0	0
$891$	59.8821	2.00613
$892$	0	0
$893$	5.37612	0.179905
$894$	0	0
$895$	13.6789	0.457234
$896$	0	0
$897$	1.61629	0.0539663
$898$	0	0
$899$	3.98870	0.133031
$900$	0	0
$901$	−67.6156	−2.25260
$902$	0	0
$903$	−11.6360	−0.387220
$904$	0	0
$905$	6.68979	0.222376
$906$	0	0
$907$	8.64992	0.287216	0.143608	−	0.989635i	$-0.454130\pi$
$907$	8.64992	0.287216	0.143608	+	0.989635i	$0.454130\pi$
$908$	0	0
$909$	−2.66997	−0.0885574
$910$	0	0
$911$	48.7347	1.61465	0.807326	−	0.590105i	$-0.200914\pi$
$911$	48.7347	1.61465	0.807326	+	0.590105i	$0.200914\pi$
$912$	0	0
$913$	19.9577	0.660504
$914$	0	0
$915$	−11.9024	−0.393481
$916$	0	0
$917$	10.4275	0.344345
$918$	0	0
$919$	6.03966	0.199230	0.0996150	−	0.995026i	$-0.468239\pi$
$919$	6.03966	0.199230	0.0996150	+	0.995026i	$0.468239\pi$
$920$	0	0
$921$	2.40811	0.0793500
$922$	0	0
$923$	17.2045	0.566292
$924$	0	0
$925$	17.9626	0.590606
$926$	0	0
$927$	1.83141	0.0601513
$928$	0	0
$929$	−3.25096	−0.106661	−0.0533303	−	0.998577i	$-0.516984\pi$
$929$	−3.25096	−0.106661	−0.0533303	+	0.998577i	$0.516984\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−2.39652	−0.0784586
$934$	0	0
$935$	−23.1874	−0.758311
$936$	0	0
$937$	−58.4338	−1.90895	−0.954474	−	0.298295i	$-0.903582\pi$
$937$	−58.4338	−1.90895	−0.954474	+	0.298295i	$0.903582\pi$
$938$	0	0
$939$	−19.9039	−0.649540
$940$	0	0
$941$	2.40089	0.0782668	0.0391334	−	0.999234i	$-0.487540\pi$
$941$	2.40089	0.0782668	0.0391334	+	0.999234i	$0.487540\pi$
$942$	0	0
$943$	3.52690	0.114852
$944$	0	0
$945$	−3.00643	−0.0977990
$946$	0	0
$947$	−17.7777	−0.577699	−0.288850	−	0.957374i	$-0.593273\pi$
$947$	−17.7777	−0.577699	−0.288850	+	0.957374i	$0.593273\pi$
$948$	0	0
$949$	5.50358	0.178654
$950$	0	0
$951$	−53.3096	−1.72868
$952$	0	0
$953$	23.5179	0.761818	0.380909	−	0.924613i	$-0.375611\pi$
$953$	23.5179	0.761818	0.380909	+	0.924613i	$0.375611\pi$
$954$	0	0
$955$	2.26504	0.0732949
$956$	0	0
$957$	−6.33765	−0.204867
$958$	0	0
$959$	6.65890	0.215027
$960$	0	0
$961$	19.2350	0.620485
$962$	0	0
$963$	−1.17389	−0.0378281
$964$	0	0
$965$	3.34320	0.107621
$966$	0	0
$967$	−11.5952	−0.372878	−0.186439	−	0.982467i	$-0.559695\pi$
$967$	−11.5952	−0.372878	−0.186439	+	0.982467i	$0.559695\pi$
$968$	0	0
$969$	−10.9234	−0.350911
$970$	0	0
$971$	27.5021	0.882586	0.441293	−	0.897363i	$-0.354520\pi$
$971$	27.5021	0.882586	0.441293	+	0.897363i	$0.354520\pi$
$972$	0	0
$973$	−19.6735	−0.630705
$974$	0	0
$975$	−11.0267	−0.353137
$976$	0	0
$977$	−51.6048	−1.65098	−0.825492	−	0.564414i	$-0.809102\pi$
$977$	−51.6048	−1.65098	−0.825492	+	0.564414i	$0.809102\pi$
$978$	0	0
$979$	8.45268	0.270149
$980$	0	0
$981$	2.29507	0.0732760
$982$	0	0
$983$	−21.4305	−0.683526	−0.341763	−	0.939786i	$-0.611024\pi$
$983$	−21.4305	−0.683526	−0.341763	+	0.939786i	$0.611024\pi$
$984$	0	0
$985$	−1.13783	−0.0362544
$986$	0	0
$987$	9.55420	0.304113
$988$	0	0
$989$	−4.45854	−0.141773
$990$	0	0
$991$	33.4365	1.06214	0.531072	−	0.847326i	$-0.321789\pi$
$991$	33.4365	1.06214	0.531072	+	0.847326i	$0.321789\pi$
$992$	0	0
$993$	64.4320	2.04469
$994$	0	0
$995$	14.0260	0.444654
$996$	0	0
$997$	53.2879	1.68764	0.843822	−	0.536623i	$-0.180300\pi$
$997$	53.2879	1.68764	0.843822	+	0.536623i	$0.180300\pi$
$998$	0	0
$999$	−19.5269	−0.617804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.ch.1.3		7
4.3	odd	2		8512.2.a.ci.1.5		7
8.3	odd	2		4256.2.a.p.1.3	✓	7
8.5	even	2		4256.2.a.q.1.5	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.p.1.3	✓	7	8.3	odd	2
4256.2.a.q.1.5	yes	7	8.5	even	2
8512.2.a.ch.1.3		7	1.1	even	1	trivial
8512.2.a.ci.1.5		7	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77715 q^{3} +0.595310 q^{5} -1.00000 q^{7} +0.158273 q^{9} +O(q^{10})\)	\(q-1.77715 q^{3} +0.595310 q^{5} -1.00000 q^{7} +0.158273 q^{9} -6.33689 q^{11} -1.33561 q^{13} -1.05796 q^{15} +6.14658 q^{17} +1.00000 q^{19} +1.77715 q^{21} +0.680950 q^{23} -4.64561 q^{25} +5.05018 q^{27} -0.562766 q^{29} -7.08767 q^{31} +11.2616 q^{33} -0.595310 q^{35} -3.86658 q^{37} +2.37358 q^{39} +5.17937 q^{41} -6.54753 q^{43} +0.0942213 q^{45} +5.37612 q^{47} +1.00000 q^{49} -10.9234 q^{51} -11.0005 q^{53} -3.77241 q^{55} -1.77715 q^{57} -6.32544 q^{59} +11.2504 q^{61} -0.158273 q^{63} -0.795101 q^{65} -10.3344 q^{67} -1.21015 q^{69} -12.8814 q^{71} -4.12066 q^{73} +8.25595 q^{75} +6.33689 q^{77} -6.21320 q^{79} -9.44977 q^{81} -3.14945 q^{83} +3.65912 q^{85} +1.00012 q^{87} -1.33389 q^{89} +1.33561 q^{91} +12.5959 q^{93} +0.595310 q^{95} -2.06937 q^{97} -1.00296 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9}+O(q^{10}) \) 7 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9	\( 7 q - 3 q^{3} + 5 q^{5} - 7 q^{7} + 8 q^{9} - 3 q^{11} + 16 q^{13} + 8 q^{15} + 8 q^{17} + 7 q^{19} + 3 q^{21} + 10 q^{23} + 8 q^{25} - 6 q^{27} + 11 q^{29} + 14 q^{31} + 3 q^{33} - 5 q^{35} + 13 q^{37} - 6 q^{39} + 11 q^{41} - 11 q^{43} - 4 q^{45} + 7 q^{47} + 7 q^{49} - 24 q^{51} + 9 q^{53} - 8 q^{55} - 3 q^{57} - 23 q^{59} + 23 q^{61} - 8 q^{63} + 40 q^{65} - 16 q^{67} + 10 q^{69} + 3 q^{71} + 4 q^{73} - 48 q^{75} + 3 q^{77} + 29 q^{79} + 23 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} + 15 q^{89} - 16 q^{91} - 42 q^{93} + 5 q^{95} - 13 q^{97} - 47 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 + 5 * q^5 - 7 * q^7 + 8 * q^9 - 3 * q^11 + 16 * q^13 + 8 * q^15 + 8 * q^17 + 7 * q^19 + 3 * q^21 + 10 * q^23 + 8 * q^25 - 6 * q^27 + 11 * q^29 + 14 * q^31 + 3 * q^33 - 5 * q^35 + 13 * q^37 - 6 * q^39 + 11 * q^41 - 11 * q^43 - 4 * q^45 + 7 * q^47 + 7 * q^49 - 24 * q^51 + 9 * q^53 - 8 * q^55 - 3 * q^57 - 23 * q^59 + 23 * q^61 - 8 * q^63 + 40 * q^65 - 16 * q^67 + 10 * q^69 + 3 * q^71 + 4 * q^73 - 48 * q^75 + 3 * q^77 + 29 * q^79 + 23 * q^81 - 6 * q^83 + 6 * q^85 - 6 * q^87 + 15 * q^89 - 16 * q^91 - 42 * q^93 + 5 * q^95 - 13 * q^97 - 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more